ZURICH—As a yearslong U.S. tax-evasion case dragged on in recent months, board members at Credit Suisse Group AG considered actions that likely would have cost Brady Dougan his job as chief executive, according to people familiar with the deliberations.

To expedite a legal settlement, Credit Suisse officials considered asking the Swiss government to implement an emergency law enabling the lender to sidestep bank-secrecy rules and hand over the names of American clients to U.S. authorities, these people said. Board members...